Bounds on 4D Conformal and Superconformal Field Theories

TL;DR

This paper establishes theoretical bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and superconformal field theories, providing tools to test the consistency of such theories.

Contribution

It derives new bounds on key parameters in 4D conformal and N=1 superconformal theories using crossing symmetry and superconformal blocks.

Findings

Lower bounds on central charge c as a function of scalar dimension d.

Bounds on current two-point function coefficients tau^{IJ} and flavor charges.

Upper bounds on the dimension of composite operators in superconformal theories.

Abstract

We derive general bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and N=1 superconformal field theories. In any CFT containing a scalar primary phi of dimension d we show that crossing symmetry of <phi phi phi phi> implies a completely general lower bound on the central charge c >= f_c(d). Similarly, in CFTs containing a complex scalar charged under global symmetries, we bound a combination of symmetry current two-point function coefficients tau^{IJ} and flavor charges. We extend these bounds to N=1 superconformal theories by deriving the superconformal block expansions for four-point functions of a chiral superfield Phi and its conjugate. In this case we derive bounds on the OPE coefficients of scalar operators appearing in the Phi x Phi* OPE, and show that there is an upper bound on the dimension of Phi* Phi when dim(Phi) is close to 1. We also present even more stringent bounds on c and tau^{IJ}. In supersymmetric gauge theories believed to flow to superconformal fixed points one can use anomaly matching to explicitly check whether these bounds are satisfied.